Linearized instability for differential equations with dependence on the past derivative

Lani-Wayda, Bernhard and Godoy Mesquita, Jaqueline (2023) Linearized instability for differential equations with dependence on the past derivative. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (52). pp. 1-52. ISSN 1417-3875

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Abstract

We provide a criterion for instability of equilibria of equations in the form ˙ x(t) = g(x′ t,xt), which includes neutral delay equations with state-dependent delay. The criterion is based on a lower bound ∆ > 0 for the delay in the neutral terms, on regularity assumptions of the functions in the equation, and on spectral assumptions on a semigroup used for approximation. The spectral conditions can be verified studying the associated characteristic equation. Estimates in the C1-norm, a manifold containing the state space X2 of the equation and another manifold contained in X2, and an invariant cone method are used for the proof. We also give mostly self-contained proofs for the necessary prerequisites from the constant delay case, and conclude with an application to a mechanical example.

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